How can I create sets of orthogonal rings on a sphere that I can extrude inward?
Use Booleans

I can't think of a topology that has the loops that you describe ready to work with.
So, depending on your project needs, I would advice to try other ways of modeling your object.

For example you could take advantage of booleans in the exact way you described.

In the main object extrude inwards the edgeloops of a sphere, then duplicate and separate those faces in another object.



Rotate the object on the Y axis by 90°, re-extrude the faces to build a watertight mesh and use it as target for a Boolean modifier to carve the grooves in the other direction while maintaining a certain degree of felexibilty in the modeling



Final result: